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OnQ-order andR-order of convergence

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Abstract

We give sufficient conditions for a sequence to have theQ-order and/or theR-order of convergence greater than one. If an additional condition is satisfied, then the sequence has an exactQ-order of convergence. We show that our results are sharp and we compare them with older results.

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References

  1. Potra, F. A., andPták, V.,Nondiscrete Induction and Iterative Processes, Pitman, Boston, Massachusetts, 1984.

    Google Scholar 

  2. Dennis, J. E.,On Newton-Like Methods, Numerische Mathematik, Vol. 11, pp. 324–330, 1967.

    Google Scholar 

  3. Wall, D.,The Order of an Iteration Formula, Mathematics of Computation, Vol. 10, pp. 167–168, 1956.

    Google Scholar 

  4. Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.

    Google Scholar 

  5. Brent, R. P.,Some Efficient Algorithms for Solving Systems of Nonlinear Equations, SIAM Journal on Numerical Analysis, Vol. 10, pp. 327–344, 1973.

    Google Scholar 

  6. Brezinski, C.,Comparaison de Suites Convergentes, Revue Française d'Informatique et de Recherche Operationelle, Vol. 2, pp. 95–99, 1971.

    Google Scholar 

  7. Schwetlick, H.,Numerische Lösung Nichtlinearer Gleichungen, VEB, Berlin, Germany, 1979.

    Google Scholar 

  8. Ostrowski, A. M.,Solution of Equations in Euclidean and Banach Spaces, 3rd Edition, Academic Press, New York, New York, 1970.

    Google Scholar 

  9. Feldstein, M. A., andFirestone, R. M.,Hermite Interpolatory Theory and Parallel Numerical Analysis, Division of Applied Mathematics, Brown University, Providence, Rhode Island, 1967.

    Google Scholar 

  10. Herzberger, J.,Über Matrixdarstellungen für Iterationsverfahren bei Nichtlinearen Gleichungen, Computing, Vol. 12, pp. 215–222, 1974.

    Google Scholar 

  11. Feldstein, M. A., andTraub, J. F.,Asymptotic Behavior of Vector Recurrences with Applications, Mathematics of Computation, Vol. 31, pp. 180–192, 1971.

    Google Scholar 

  12. Schmidt, J. W.,On the R-Order of Coupled Sequences, Computing, Vol. 26, pp. 333–342, 1981.

    Google Scholar 

  13. Burmeister, W., andSchmidt, J. W.,On the R-Order of Coupled Sequences, Parts II and III, Computing, Vol. 29, pp. 73–81, 1982 and Vol. 30, pp. 157–169, 1984.

    Google Scholar 

  14. Tapia, R. A., andWhitley, D.,On the Superquadratic Convergence of Projected Newton Method for the Symmetric Eigenvalue Problem, Technical Report No. 87-14, Department of Mathematical Sciences, Rice University, Houston, Texas, 1987.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Iowa, Iowa City, Iowa

    F. A. Potra (Associate Professor)

Authors
  1. F. A. Potra
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Communicated by R. A. Tapia

This work was supported in part by the National Science Foundation under Grant No. DMS-85-03365. The author wishes to thank J. E. Dennis and R. A. Tapia for helpful comments, and the referee for pointing out a number of typographical and mathematical errors in the original version of this paper.

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Potra, F.A. OnQ-order andR-order of convergence. J Optim Theory Appl 63, 415–431 (1989). https://doi.org/10.1007/BF00939805

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